Dedekind--Kummer prime factorisation

Put $A={\mathcal{O}}_K$, $B={\mathcal{O}}_L$ and $k=A\mathrm{/}\mathfrak{p}$. Monic division by $F$ shows that the kernel of $A[X]\to B$, $X\mapsto \alpha$, is $(F)$: a remainder of degree less than $\mathrm{deg}F$ vanishing at $\alpha$ is zero by minimality over $K$. Monogeneity gives surjectivity, hence $B\mathrm{/}\mathfrak{p}B\simeq k[X]\mathrm{/}(\bar{F})$.

The maximal ideals of this quotient are exactly those generated by the ${\bar{g}}_i$; their inverse images in $B$ are ${\mathfrak{P}}_i=(\mathfrak{p},g_i(\alpha ))$. They are independent of the chosen lifts. Their residue fields are $k[X]\mathrm{/}({\bar{g}}_i)$, of degree $\mathrm{deg}{\bar{g}}_i$, and their contractions are $\mathfrak{p}$. Thus they are exactly the primes above $\mathfrak{p}$.

The local factorisation theorem Integral ideal factorisation in a number field, in ZF writes $\mathfrak{p}B={\prod }_i{\mathfrak{P}}_i^{a_i}$. In the DVR $B_{{\mathfrak{P}}_i}$ used in that theorem's proof, the maximal ideal of $B_{{\mathfrak{P}}_i}\mathrm{/}\mathfrak{p}{B}_{{\mathfrak{P}}_i}$ has nilpotency index exactly $a_i$. On the polynomial side of step 1.1, localisation at $({\bar{g}}_i)$ gives $k[X{]}_{({\bar{g}}_i)}\mathrm{/}({\bar{g}}_i^{e_i})$, whose maximal ideal has nilpotency index exactly $e_i$: its $e_i$-th power vanishes and its $(e_i-1)$-st power does not. Thus $a_i=e_i$, proving the ideal factorisation and the stated ramification and residue degrees.